Abstract:Continuous causal discovery typically couples representation learning with structural optimization via non-convex acyclicity penalties, which subjects solvers to local optima and restricts scalability in high-dimensional regimes. We propose a decoupled paradigm that shifts the causal discovery bottleneck from non-convex optimization to statistical score estimation. We introduce the Score-Schur Topological Sort (SSTS), an algorithm that extracts topological order directly from unconstrained generative models, bypassing constrained structure optimization. We establish that the causal hierarchy leaves a geometric signature within the score function: iterative graph marginalization is mathematically equivalent to computing the Schur complement of the Score-Jacobian Information Matrix (SJIM) under linear conditions. This translates the acyclicity constraint into an algebraic procedure with a dominant cost of O(d^3) operations. For non-linear systems, we formulate the expectation gap of Schur marginalization and introduce Block-SSTS to compress extraction depth, bounding structural error. Empirically, SSTS allows causal structural analysis on non-linear graphs up to d=1000. At this scale, our framework indicates that once the non-convex optimization bottleneck is mathematically bypassed, the structural fidelity of continuous causal discovery is bounded by the finite-sample estimation variance of the global score geometry. By reducing graph extraction to matrix operations, this work reframes scalable causal discovery from a constrained optimization problem to a statistical estimation challenge.
| Comments: | 18 pages, 3 figures, 7 tables |
| Subjects: | Machine Learning (cs.LG) |
| MSC classes: | 68T05, 62H22 |
| ACM classes: | I.2.6; G.3 |
| Cite as: | arXiv:2604.25295 [cs.LG] |
| (or arXiv:2604.25295v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2604.25295 arXiv-issued DOI via DataCite (pending registration) |
From: Rui Wu [view email]
[v1]
Tue, 28 Apr 2026 07:03:04 UTC (84 KB)
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